With data samples collected with the BESIII detector operating at the BEPCII storage ring at center-of-mass energies from 4.009 to 4.420 GeV, the process e⁺e⁻ → γ X(3872) is observed for the first time with a statistical significance of 6.3σ. The measured mass of the X(3872) is (3871.9 ± 0.7(stat) ± 0.2(syst)) MeV/c², in agreement with previous measurements. Measurements of the product of the cross section σ[e⁺e⁻ → γ X(3872)] and the branching fraction ℬ [X(3872) → π⁺π⁻J/ψ] at center-of-mass energies 4.009, 4.229, 4.260, and 4.360 GeV are reported. Our measurements are consistent with expectations for the radiative transition process Y(4260) → γ X(3872).